The class of grim reapers in $\mathbb{H}^2\times\mathbb{R}$

Antonio Bueno; Rafael López

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arXiv · 2402.05772

The class of grim reapers in $\mathbb{H}^2\times\mathbb{R}$

Antonio Bueno·Rafael López

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Abstract

We study translators of the mean curvature flow in the product space $\h^2\times\r$. In $\h^2\times\r$ there are three types of translations: vertical translations due to the factor $\r$ and parabolic and hyperbolic translations from $\h^2$. A grim reaper in $\h^2\times\r$ is a translator invariant by a one-parameter group of translations. The variety of translators and translations in $\h^2\times\r$ makes that the family of grim reapers particularly rich. In this paper we give a full classification of the grim reapers of $\h^2\times\r$ with a description of their geometric properties. In some cases, we obtain explicit parametrizations of the surfaces.

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Antonio Bueno, Rafael López. 2024-02-08. The class of grim reapers in $\mathbb{H}^2\times\mathbb{R}$. https://arxiv.org/abs/2402.05772

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